An Integrability Criterion for Banach-Lie Triple Systems

TL;DR

This paper establishes a criterion for the integrability of Banach-Lie triple systems by linking it to the discreteness of their period group, extending concepts from Lie algebra theory to symmetric spaces.

Contribution

It introduces the period group for Banach-Lie triple systems and proves that integrability is equivalent to this group being discrete, generalizing existing Lie algebra results.

Findings

A Lie triple system is integrable iff its period group is discrete.

Constructed the period group for Banach-Lie triple systems.

Extended the path and loop space constructions to symmetric spaces.

Abstract

To give a criterion for the integrability of Banach-Lie triple systems, we follow the construction of the period group of a Lie algebra and define the period group of a Lie triple system as an analogous concept. We show that a Lie triple system is integrable if and only if its period group is discrete. Along the way, we see how to turn the path and the loop space of a pointed symmetric space into pointed symmetric spaces.